Different defaults

Never assume that other browsers will have the same settings by default as your own.

The most obvious examples are the features the user can change in the browser's preference settings: font face, font size, foreground and background colours.
If you want your web page to look identical across different browsers, set all these explicitly in your stylesheet.
Or, if you want to respect the user's preferences, don't assume that those preferences are the same as your own!

But those are not the only differences. There are other cases where browsers have different defaults, and where you can either respect those differences by not assuming anything, or specify the offending properties in your stylesheet.
There are many "reset stylesheets" to be found on the web, that reset properties of all elements to the same value, but not only are those overkill - they make your document look like plain text, really - they also don't address all of these specific differences:

Other differences, per browser. Note: the browser names are rough groupings, mouse over them to see details

HR properties
Browser colour height on screen top/bottom margin with noshade
Chrome color attr or border-color style same as SIZE 1/2 em (*) bg is background-color or color attr
Firefox border-color style or color style or color attr same as SIZE 1/2 em rounded corners; bg is border-color style or color style or color attr
IE border-color style or black if a color attr is present equal to SIZE+2 1/2 em bg is background-color style or black if a color attr is present
Opera color attr or border-color style or color style same as SIZE 1/2 em (*) bg is background-color or color attr
Konqueror border-color style or color attr same as SIZE 12 pixels bg is border-color style or color attr

(*) Opera and Safari round the values down (if 1em=15px, the margin will be 7px). Other browsers keep the total margin at 1em (so top=7px and bottom=8px).
In other words, if you want a hr to look the same across browsers, set all the styles explicitly; don't rely on the defaults. Example:
hr {color:#888; border:1px solid; border-color:#555 #888 #bbb; box-sizing:content-box; margin:.5em 0;}
hr[noshade] {background:#888; border-radius:0; border:1px solid #888;}

Appearance of various elements
Browser dfn abbr, acronym textarea listing xmp plaintext
Chrome italic normal text 13px font block, margins block, margins block, margins
Firefox italic dotted underline monospace font, size block, margins block, margins block, margins
IE italic dotted underline 13px font inline, smaller font inline block
Opera italic dotted underline ? block block, margins block
Konqueror normal small caps ? block, margins block, margins block, margins
Appearance of various styles
Browser font-family: monospace
Chrome font from settings; size 3px less than the parent
Firefox font and size from settings
IE Chooses its own font
Opera font from settings, keeps size
Konqueror font and size from settings

Note: IE7 and IE8 let the user choose which font to use for fixed-width fonts. However, this font is only used for elements like <pre>, <kbd>, <tt> and so on. For style="font-family: monospace" or <font face="monospace"> IE disregards the user preferences and chooses a font itself. Also the relative size seems to be different.

Differences in font weight handling
Browser "Light" fonts
max weight¹
"Bold" fonts
min weight
style of <b> effect of "lighter"effect of "bolder"
Chrome n.a.700bold(seems to depend on font)
Firefox 300600boldersubtract 200bolder variant
IE 300600bolderlighter variantbolder variant
Opera n.a.600boldersubtract 200add 200
Konqueror n.a.600boldsame as normalsame as bold

¹ Some fonts, like DejaVu Sans, have three weights: light, regular and bold; this column shows the largest font-weight that comes out light. Not all browsers support this; the column says "n.a." in those cases.

The box-sizing model for all elements is supposed to be content-box. That is, when calculating sizes, the content area is supposed to be the size of the width or height property, not counting the border. However, many browsers apply the border-box model on some random elements, that is, they use the width or height properties for the outer size of the element, making the content area smaller.

Browserhrtd, thinput
Chrome border-box¹
Firefox border-boxn.a.²
IE border-box
Opera border-box³border-boxborder-box
Konqueror border-box³n.a.²n.a.²

¹ Doesn't actually draw the border though.
² n.a. means the element doesn't respond to this style.
³ Only the height; the width is calculated according to the standard!

Differences in size for <small> and <big>
Browser <small><big>
Chrome 83.3333%120%
Firefox <=52 66.6667%150%
current Firefox 83.3333%120%
IE t.b.d.t.b.a
Opera t.b.d.t.b.a
Konqueror 83.3333%120%
Margins inside nested lists, i.e. a list in a <li> in a <ul>
Browser <ul> or <ol><menu><dl>
Chrome suppressed marginsno effectno effect
Firefox suppressed marginssuppressed marginssuppressed margins
IE newline beforenewline beforenewline before
Opera suppressed marginsno effectsuppressed margins
Konqueror suppressed marginssuppressed marginsno effect